1. Field of Invention
The present invention generally relates to the metrology of three-dimensional surfaces and, more particularly, relates to a method and apparatus for describing surface characteristics such as surface area, surface area enhancement, and surface roughness parameters, based solely on line-based measurements and without requiring the calculations to be derived from, or based on, a three-dimensional image of the surface, or on data necessary to generate such an image.
2. Description of Related Art
A real surface may be generally defined as the infinitesimally thin outer skin of a three-dimensional object. Thus, while the three-dimensional object itself has an associated appreciable volume, the surface of the three-dimensional object does not. Likewise, a single, generically shaped line-trace over the three-dimensional surface does not have any characteristics of area, but rather has the one-dimensional characteristic of length.
With regard to spatial directions on a surface, the Cartesian-coordinate system is one acceptable way to reference relative positions of general locations on that surface. In general, the orientation of the entire coordinate system is arbitrarily positioned with regard to a surface and its features, or components. In the case of a surface-metrology instrument, the Cartesian coordinate system axes are usually fixed by the design of the instrument and are specified according to this frame of reference. Generally, the determination of surface characteristics such as the actual surface area of a non-planar surface requires a full consideration of all three spatial dimensions, namely width (X-direction), depth (Y-direction), and height (Z-direction).
A surface metrology technique based on all three dimensions requires use of a metrology tool that is capable of accurately tracking the surface topography and recording relative Z heights with respect to both the X position and the Y position of a particular Z height measurement. The manner in which the data are thus acquired and manipulated to determine surface characteristics distinguishes related art from the present invention.
The related art generates a topographical map of the surface of the sample from which the sample""s surface area may be determined. The map may either be recorded in analog or digital fashion. In the case of a digital measurement, the representative map typically consists of a two-dimensional array of data points or xe2x80x9cpixelsxe2x80x9d with an associated measured Z height for each X, Y position on the map.
One important characterization task for which the invention is useful is the determination of an actual total measured surface area (xe2x80x9cAACTxe2x80x9d) from data obtained while scanning a surface. One accepted method for determining AACT from a digitized topographical map of a non-planar surface is to describe the surface area as an array of n-subareas or shapes and to sum all of the areas of the n-shapes. The area of each of the n-shapes is the geometric area of each shape (xe2x80x9canxe2x80x9d), formed between adjacent Z height values, where AACT=xcexa3an. The area of each shape is calculated from the relative height values of the adjoining pixels and basic geometric principles. The resulting surface area of each shape is counted once toward AACT.
It is common practice to also describe a measured surface in terms of a xe2x80x9cnormalizedxe2x80x9d surface area. The normalized surface area, or xe2x80x9csurface area enhancementxe2x80x9d (xe2x80x9cSAExe2x80x9d), is the ratio of the measured surface area, as for example AACT described above, to the projected surface area (xe2x80x9cAPROJxe2x80x9d). APROJ is an area corresponding to the projection of the three-dimensional surface AACT onto an X-Y plane. APROJ is defined completely by the extrema of the X and Y coordinates of the topographical map. This ratio is bounded at the lower limit by unity, for a perfectly flat and level surface, and theoretically has no upper limit. The practical limitation on the ability to completely track the real surface, and to thereby accurately determine the SAE, is determined by the method employed to acquire the topographical map and is not to be considered a limiting factor in the scope of the present invention.
The related art for the determination of SAE is primarily limited in accuracy by the relative correspondence of the recorded Z height values to the actual topography of the surface. A secondary limiting factor in the accuracy of SAE determination and other surface characteristic determinations is the spatial frequency at which, or detail with which, the actual surface is sampled. In the example of a digitized topographical surface map, the sampling frequency corresponds to the spatial frequency of the pixels. Such a xe2x80x9cgridxe2x80x9d of pixels is usually arranged with equal spacing in both the X and Y directions and is typically constructed in a line-by-line fashion. Each such line is m pixels in length, and there are n lines. The result is the tracing of a grid of n lines by m pixels which extend in the X direction, and which are spaced from one another in the Y direction.
The primary consideration in determining the appropriate grid dimensions is that each surface feature is sampled in two dimensions sufficiently for the related art methodology to provide an accurate estimation of the surface area. Instruments capable of performing these functions are limited to scanning probe microscopes (SPMs), including atomic force microscopes (AFMs) and other instruments capable of acquiring and/or manipulating data usable to generate a three-dimensional image of the topography of a surface. Profilometers, which merely ascertain the profile of a two-dimensional line extending across this surface of the sample, generally cannot generate such an image without some additional means to raster the stylus, or surface to be characterized, in both X and Y dimensions while recording Z height values associated with a particular location on that surface.
Those skilled in the art of metrology and instrumentation recognize that the accuracy of a surface area measurement, a SAE measurement, or another surface characteristic measurement according to the related art could be greatly improved by increasing the spatial sampling rate in both the X and Y directions. However, one skilled in the art will also recognize that such an increase in the spatial sampling rate usually requires increased measurement time.
Furthermore, a skilled practitioner of surface metrology techniques is aware of the sensitivity of the related art methodology to measurement noise, particularly where such noise results from an artifactual mis-registration or misalignment between pixels in adjacent lines of pixels. This type of noise is typically referred to as xe2x80x9clow frequency noisexe2x80x9d because it is superimposed on the topographical map at rates much lower than the usual rate of measurement. It is also sometimes referred to as xe2x80x9cinter-line noisexe2x80x9d because it reflects differences in measurement along two adjacent traces in the X direction. Such inter-line or low frequency noise is particularly pronounced in related art SAE measurements because the noise in those measurements is shifted into the frequency domain at which the actual height data is recorded. Any relative displacement in X, Y, or Z data between entire lines will necessarily result in a less accurate measurement of actual surface area and SAE. Any such displacement resulting from various sources of low frequency noise contaminates the data from image-based surface metrology related art methods.
In order to minimize both measurement time and sensitivity to low frequency noise, a novel, line-based measurement method to determine SAE and other surface characteristics is proposed.
The present invention has been developed to overcome the above-described limitations in existing methodology for the determination of one or more surface characteristics of a three-dimensional surface.
It is accordingly an object of the present invention to provide an improved method for determining one or more surface characteristics of a three-dimensional surface.
Another object of the present invention is to provide an improved method for measuring the surface area of a three-dimensional surface.
Still another object is to provide an improved method of determining SAE.
In accomplishing these and other objects, the present invention provides for measuring the one-dimensional xe2x80x9csurfacexe2x80x9d length of an arbitrarily shaped line traced over a surface of a three-dimensional object in order to determine at least one surface characteristic of interest. A method of making such a measurement according to the present invention comprises the steps of 1) physically measuring and reporting the total surface length of at least one line traced over at least a given region of the surface of the three-dimensional object, and 2) determining the surface characteristic of interest using the traced line(s).
In the case of an SAE measurement, the invention additionally includes determining and reporting the total projected length of the traced line(s), and finding the surface area enhancement of a given region associated with the line(s) as the squared ratio of the sum of individual surface lengths to the sum of individual projected lengths of the line(s).
As a result of the line-scan derivation of the surface area enhancement, the measurement method of the present invention is intrinsically insensitive to the low frequency noise problems of related art image-based surface area measurements. Another advantage of the present invention is an improvement in measurement speed over image-based measurements. A third benefit that applies to certain methods of tracing lines is the possible improved lifetime of the instrument. More specifically, probe-based and other physically-based surface metrology instruments often wear or break due to wear and tear occurring during data acquisition. The potential for a statistically justified reduction in the number of line traces required to make a reasonable estimate of the surface area can result in a proportionate increase in the lifetime of the measuring apparatus.